Why Does Kakashi Kill Rin?
Have you ever noticed our prodigy of Konoha being branded with the nickname as "The Friend Killer Kakashi." This happened right after the end of the fated mission wherein he lost two if his friends, Rin Nohara, and Obito Uchiha.
Clearing things up, Kakashi Hatake did not kill Rin Nohara. Rather, she commited suicide by impaling herself on Kakashi's Chidori which was meant to hit an enemy shinobi, dying on the hands of the boy she likes and sacrificing herself for the greater good of her beloved village.
Everyone could see Kakashi as a lazy and laid back Shinobi. He is also very well known for his dislike towards responsibilities. This is not because of his personality, rather he is well-known to be a genius prodigy since young. He acts like this due to the emotional torment he experiences from the friends he has lost. This is why after forming Team 7 composed of Naruto Uzumaki, Sasuke Uchiha, and Sakura Haruno, he stressed about how important Teamwork is. Since this was a painful lesson he received together with the nicknames of Friend Killer and Cold-Blooded Kakashi!
The Real Reason Why Did Kakashi Kill Rin
Rin Nohara was kidnapped by Madara Uchiha, under the guise of Kirigakure and was forcibly turned into a jinchuriki of the Three-Tailed Beast known as Isobu. This was a terrible scheme to destroy Konohagakure, one of the most striving Villages of the Shinobi World. It doesn't end there as Rin was actually given a curse that when she sets foot into Konoha, the tailed-beast within her will let loose and rampage killing thousands of innocent lives.
Kakashi Hatake saved him from her predicament, but that didn't last long as various elite ninjas was sent after them. The duo escaped with the Kiri-nin hot within their tails but ultimately failed as they couldn't fully outrun the enemies, ending on with Kakashi and Rin facing off the Shinobis of Kirigakure that was sent after them.
Rin as a jinchuuriki was a sin itself, especially in times of war in the Third Great Ninja War. Which led to many powers coveting such strength. She did not want to become a tool to destroy her beloved village nor does she want to hurt anyone so she begged Kakashi to kill her but failed as he promised to Obito to protect Rin Nohara. She eventually forces herself in kakashi's powerful strike taking in the full brunt of the damage and killing herself in the process, smiling at Kakashi in the end. Sadly such heroic deed was not uncovered only after a the middle part of Naruto Shippuden.Series.
Kakashi Kills Rin Episode
Kakashi Kills Rin Episode transpired on a flashback of Obito Uchiha at episode 345 of Naruto Shippuden. This very event as well activated both Kakashi and Obito's Mangekyo Sharingan. The Third Shinobi World War took a lot of casualties from all the sides be it enemy or foe. And for Konohagakure, Rin's Death was something that not only caused grief but also sparked the Fourth Shinobi World War. Of course this did not happen right after Rin's death, rather it was weaved by Madara Uchiha. An inescapable scheme that will haunt Kakashi Hatake in the future, not by memories, but by Obito Uchiha in the flesh.
Kakashi After Rin's Death
Death was totally not that new to Kakashi as his mother dying while he was very young, and his father died through committing suicide. This left a deep scar on our shinobi and lived a life strictly following the Ninja's Code. This was also the very reason he was built differently from his other peers and matured early. Having lost both Obito Uchiha, and failed on protecting Rin Nohara, even becoming one of the main proprietor of her death, albeit it being accidental, this did not let Kakashi falling into depression.
Everyone around him at that time even believed that he will kill anyone, be it friend or foe, enstranging him even further to the people he was once close to. Minato Namikaze, as their sensei, helped Kakashi in a way by transferring him to protect Kushina Uzumaki. Slowly Kakashi Hatake was healed by those who persevered to break the ice cold heart, and visited Rin and Obito's grave during his time-off. Albeit his slowly healing wound, he still suffers post-traumatic stress disorder, reliving the day he killed Rin by his hands.